Subgroups — #28745114

Order 15104.

488 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434755	C₂	1	1	yes	maximal, characteristic, complemented
2	#21434746	C₂	1	1	yes	maximal, characteristic
3	#21426261	C₂	1	2	yes	maximal, complemented
4	#21426254	C₂	1	2	yes	maximal, complemented
5	#21426209	C₂	1	2	yes	maximal, complemented
6	#21426192	C₂	1	2	yes	maximal, complemented
7	#21426142	C₂	1	2	yes	maximal, complemented
8	#21426137	C₂	1	2	yes	maximal, complemented
9	C₅₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	#16838014	—	2	1	no	—
11	#16838007	—	2	1	no	—
12	C₂×#14273672	C₂²	1	2	yes	—
13	C₂×#14273672	C₂²	1	2	yes	—
14	#16837804	C₂²	1	1	yes	characteristic
15	#16837804	C₂²	1	1	yes	characteristic
16	#16837782	C₂²	1	1	yes	characteristic
17	#16837779	C₂²	1	1	yes	characteristic
18	#16837771	—	2	1	no	—
19	#16837767	—	2	1	no	—
20	#16837660	—	2	1	no	—
21	#16837655	—	2	1	no	—
22	#16837643	C₂²	1	2	yes	—
23	#16837638	C₂²	1	2	yes	—
24	#16837515	—	2	1	no	—
25	#16837510	—	2	1	no	—
26	C₄×Dic₂₃₆	C₂²	1	2	yes	—
27	C₄×D₄₇₂	C₂²	1	2	yes	—
28	#16837230	C₂²	1	2	yes	—
29	#16837228	C₂²	1	2	yes	—
30	#16837207	C₂²	1	2	yes	—
31	#16837207	C₂²	1	2	yes	—
32	#16837204	C₂²	1	2	yes	—
33	#16837204	C₂²	1	2	yes	—
34	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
36	C₅₉×#405	C₂²	1	1	yes	characteristic
37	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
38	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₂×Dic₂₃₆	—	2	1	no	—
40	C₂×Dic₂₃₆	C₂³	1	2	yes	—
41	C₂×D₄₇₂	—	2	1	no	—
42	C₂×D₄₇₂	C₂³	1	2	yes	—
43	#14273727	—	2	1	no	—
44	#14273724	—	2	1	no	—
45	#14273720	—	4	2	no	—
46	#14273719	—	2	1	no	—
47	#14273719	—	2	1	no	—
48	#14273717	—	4	2	no	—
49	#14273712	—	4	2	no	—
50	#14273706	—	2	2	no	—
51	#14273701	—	2	2	no	—
52	#14273700	—	4	2	no	—
53	C₄×Dic₁₁₈	—	2	2	no	—
54	C₄×D₂₃₆	—	2	2	no	—
55	#14273672	—	4	2	no	—
56	#14273672	—	4	2	no	—
57	#14273661	C₂³	1	2	yes	—
58	#14273661	C₂³	1	2	yes	—
59	#14273659	—	2	1	no	—
60	#14273659	—	2	1	no	—
61	#14273652	—	2	2	no	—
62	#14273652	—	2	2	no	—
63	#14273649	—	2	2	no	—
64	#14273649	—	2	2	no	—
65	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	#117×C₅₉	D₄	1	1	yes	characteristic
68	M₃₂×C₅₉	—	4	1	no	—
69	C₂×C₁₆×C₅₉	—	4	1	no	—
70	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
71	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
72	#97×C₅₉	D₄	1	1	yes	characteristic
73	#97×C₅₉	D₄	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₅₉	—	4	2	no	—
78	C₂×D₂₃₆	—	2	2	no	—
79	C₂×D₂₃₆	—	4	2	no	—
80	C₂×C₄×D₅₉	—	4	2	no	—
81	C₂×Dic₁₁₈	—	2	2	no	—
82	M₁₆×C₅₉	—	4	1	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	—	2	1	no	—
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
90	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
91	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂²⋊Dic₅₉	—	4	2	no	—
94	(C₂×D₅₉)⋊C₄	—	4	2	no	—
95	(C₂×D₅₉)⋊C₄	—	4	2	no	—
96	C₄⋊Dic₅₉	—	2	2	no	—
97	C₄⋊Dic₅₉	—	2	2	no	—
98	C₄⋊Dic₅₉	—	4	2	no	—
99	Dic₅₉⋊C₄	—	4	2	no	—
100	Dic₅₉⋊C₄	—	4	2	no	—
101	Dic₅₉⋊C₄	—	4	2	no	—
102	Dic₅₉⋊C₄	—	4	2	no	—
103	C₄×Dic₅₉	—	4	2	no	—
104	Dic₂₃₆	—	2	2	no	—
105	Dic₂₃₆	—	2	2	no	—
106	Dic₂₃₆	—	4	2	no	—
107	D₄₇₂	—	2	2	no	—
108	D₄₇₂	—	2	2	no	—
109	D₄₇₂	—	4	2	no	—
110	C₁₆×C₅₉	—	2	2	no	—
111	C₁₆×C₅₉	—	2	2	no	—
112	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₅₉	—	4	2	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
120	C₂²⋊D₅₉	—	16	2	no	—
121	C₂×Dic₅₉	—	4	2	no	—
122	C₂×Dic₅₉	—	4	2	no	—
123	C₂×Dic₅₉	—	4	2	no	—
124	D₂₃₆	—	4	2	no	—
125	D₂₃₆	—	4	2	no	—
126	D₂₃₆	—	16	2	no	—
127	C₄×D₅₉	—	16	2	no	—
128	Dic₁₁₈	—	4	2	no	—
129	Dic₁₁₈	—	4	2	no	—
130	C₈×C₅₉	—	2	1	no	—
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	—	2	1	no	—
133	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
135	#33826	—	59	1	no	maximal, Sylow
136	C₂²×C₅₉	—	2	2	no	—
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
139	C₂×D₅₉	—	8	2	no	—
140	C₂×D₅₉	—	8	2	no	—
141	C₂×D₅₉	—	8	2	no	—
142	C₄×C₅₉	—	2	2	no	—
143	C₄×C₅₉	—	2	2	no	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	#537	1	1	yes	characteristic
146	C₄×C₅₉	#575	1	1	yes	characteristic
147	Dic₅₉	—	8	2	no	—
148	Dic₅₉	—	8	2	no	—
149	Dic₅₉	—	8	2	no	—
150	#3177	—	59	1	no	—
151	#3176	—	59	1	no	—
152	#2229	—	59	2	no	—
153	#2228	—	59	2	no	—
154	#2220	—	59	2	no	—
155	#2217	—	59	2	no	—
156	#2208	—	59	2	no	—
157	#2207	—	59	2	no	—
158	#2151	D₅₉	1	1	yes	characteristic, complemented
159	C₂×C₅₉	—	4	1	no	—
160	C₂×C₅₉	#3245	1	1	yes	characteristic
161	C₂×C₅₉	#3416	1	2	yes	—
162	D₅₉	—	16	2	no	—
163	#541	—	118	1	no	—
164	#540	—	118	1	no	—
165	C₂×SD₃₂	—	59	2	no	—
166	C₂×SD₃₂	—	59	2	no	—
167	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
168	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
169	#499	—	59	1	no	—
170	#499	—	59	1	no	—
171	#494	—	59	1	no	—
172	#493	—	59	1	no	—
173	#491	—	118	1	no	—
174	#490	—	118	1	no	—
175	#470	—	118	1	no	—
176	#469	—	118	1	no	—
177	#467	—	59	2	no	—
178	#466	—	59	2	no	—
179	#442	—	118	1	no	—
180	#441	—	118	1	no	—
181	C₄×Q₁₆	—	59	2	no	—
182	C₄×D₈	—	59	2	no	—
183	#405	C₂×D₅₉	1	1	yes	characteristic
184	#367	—	59	2	no	—
185	#366	—	59	2	no	—
186	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
187	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
188	#358	—	59	2	no	—
189	#358	—	59	2	no	—
190	#357	—	59	2	no	—
191	#357	—	59	2	no	—
192	C₅₉	#33826	1	1	yes	characteristic, complemented, Sylow
193	C₂×Q₁₆	—	59	2	no	—
194	C₂×Q₁₆	—	118	1	no	—
195	C₂×D₈	—	59	2	no	—
196	C₂×D₈	—	118	1	no	—
197	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	118	1	no	—
200	#127	—	118	1	no	—
201	#126	—	236	2	no	—
202	#125	—	118	1	no	—
203	#125	—	118	1	no	—
204	#125	—	236	2	no	—
205	#123	—	236	2	no	—
206	C₂²⋊Q₈	—	118	2	no	—
207	#121	—	118	2	no	—
208	#121	—	236	2	no	—
209	C₄×Q₈	—	118	2	no	—
210	C₄×D₄	—	118	2	no	—
211	#117	D₂₃₆	1	1	yes	characteristic
212	SD₃₂	—	236	2	no	—
213	SD₃₂	—	236	2	no	—
214	M₃₂	—	4	1	no	—
215	C₂×C₁₆	—	4	1	no	—
216	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
217	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
218	#107	—	59	2	no	—
219	#107	—	59	2	no	—
220	#106	—	118	1	no	—
221	#106	—	118	1	no	—
222	#103	—	118	2	no	—
223	#103	—	118	2	no	—
224	#102	—	118	2	no	—
225	#102	—	118	2	no	—
226	#97	D₂₃₆	1	1	yes	characteristic
227	#97	D₂₃₆	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
230	C₂×Q₈	—	118	2	no	—
231	C₂×D₄	—	118	2	no	—
232	C₂×D₄	—	236	2	no	—
233	C₂×D₄	—	236	2	no	—
234	C₂²×C₄	—	236	2	no	—
235	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
236	Q₁₆	—	118	2	no	—
237	Q₁₆	—	118	2	no	—
238	Q₁₆	—	236	2	no	—
239	D₈	—	118	2	no	—
240	D₈	—	118	2	no	—
241	D₈	—	236	2	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
246	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
247	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
249	C₄⋊C₄	—	118	2	no	—
250	C₄⋊C₄	—	118	2	no	—
251	C₄⋊C₄	—	236	2	no	—
252	C₄⋊C₄	—	236	2	no	—
253	C₄⋊C₄	—	236	2	no	—
254	C₄⋊C₄	—	236	2	no	—
255	C₄⋊C₄	—	236	2	no	—
256	C₄⋊C₄	D₄₇₂	1	2	yes	—
257	C₂²⋊C₄	—	236	2	no	—
258	C₂²⋊C₄	—	236	2	no	—
259	C₂²⋊C₄	—	236	2	no	—
260	C₂²⋊C₄	D₄₇₂	1	2	yes	—
261	C₄²	—	236	2	no	—
262	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
263	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
264	C₁₆	—	2	2	no	—
265	C₁₆	—	2	2	no	—
266	C₂³	—	236	2	no	—
267	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
268	Q₈	—	236	2	no	—
269	Q₈	—	236	2	no	—
270	D₄	—	944	2	no	—
271	D₄	—	944	2	no	—
272	D₄	—	236	2	no	—
273	D₄	—	236	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	944	2	no	—
277	C₂×C₄	—	236	2	no	—
278	C₂×C₄	—	236	2	no	—
279	C₂×C₄	—	236	2	no	—
280	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
281	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
282	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	#14273772	1	1	yes	characteristic
288	C₈	#14273775	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	472	2	no	—
292	C₂²	—	472	2	no	—
293	C₂²	—	472	2	no	—
294	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	472	2	no	—
299	C₄	—	472	2	no	—
300	C₄	—	472	2	no	—
301	C₄	#16837983	1	1	yes	characteristic
302	C₄	#16838202	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	944	2	no	—
305	C₂	#21435406	1	1	yes	characteristic
306	C₂	#21437244	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2151	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#21434746	C₂
hypercenter	#2151	D₅₉
hypocenter	C₅₉	#33826
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #21434755	C₂	semidirect
3. #21426261	C₂	semidirect
4. #21426254	C₂	semidirect
5. #21426209	C₂	semidirect
6. #21426192	C₂	semidirect
7. #21426142	C₂	semidirect
8. #21426137	C₂	semidirect
9. C₅₉×#2151	C₂	semidirect
158. #2151	D₅₉	semidirect
192. C₅₉	#33826	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21434755	2
#21434746	2
#21426261	2
#21426254	2
#21426209	2
#21426192	2
#21426142	2
#21426137	2
C₅₉×#2151	2
#33826	59